A Minimax Equality Related to the Longest Directed Path in an Acyclic Graph

1975 ◽  
Vol 27 (2) ◽  
pp. 348-351 ◽  
Author(s):  
K. Vidyasankar ◽  
D. H. Younger
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Directed Path ◽  
Acyclic Graph ◽  
Vertex Set ◽  
Acyclic Directed Graph

As an analog of a recently established minimax equality for directed graphs [1], I. Simon has suggested that the following be investigated.1.1. For a finite acyclic directed graph G, a minimum collection of directed coboundaries whose union is the edge set of G has cardinality equal to that of a maximum strong matching of G.This minimax equality is here proved, using a characterization of a maximum strong matching of an acyclic graph as the set of edges of a longest directed path in the graph.The terms employed in the above theorem are defined as follows. Let G be a finite directed graph with vertex set VG and edge set eG

scholarly journals OBSTRUCTIONS TO A GENERAL CHARACTERIZATION OF GRAPH CORRESPONDENCES

2013 ◽  
Vol 95 (2) ◽  
pp. 169-188
Author(s):  
S. KALISZEWSKI ◽  
NURA PATANI ◽  
JOHN QUIGG
Keyword(s):  
Directed Graph ◽  
Discrete Space ◽  
Topological Graphs ◽  
Locally Compact ◽  
General Characterization ◽  
Product Systems ◽  
Vertex Set ◽  
Higher Rank

AbstractFor a countable discrete space $V$, every nondegenerate separable ${C}^{\ast } $-correspondence over ${c}_{0} (V)$ is isomorphic to one coming from a directed graph with vertex set $V$. In this paper we demonstrate why the analogous characterizations fail to hold for higher-rank graphs (where one considers product systems of ${C}^{\ast } $-correspondences) and for topological graphs (where $V$ is locally compact Hausdorff), and we discuss the obstructions that arise.

scholarly journals A Construction for the Hat Problem on a Directed Graph

10.37236/1994 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Rani Hod ◽  
Marcin Krzywkowski
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Maximum Clique ◽  
Clique Number ◽  
Bipartite Graphs ◽  
The Other ◽  
Complete Graphs ◽  
Undirected Graphs ◽  
Asymptotically Optimal

A team of $n$ players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph.Previous works focused on the hat problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, and bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique.We show that the conjecture does not hold for directed graphs. Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.

On Subtrees of Directed Graphs with No Path of Length Exceeding One

1970 ◽  
Vol 13 (3) ◽  
pp. 329-332 ◽  
Author(s):  
R. L. Graham
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Directed Path ◽  
Directed Tree ◽  
Prove Theorem

The following theorem was conjectured to hold by P. Erdös [1]:Theorem 1. For each finite directed tree T with no directed path of length 2, there exists a constant c(T) such that if G is any directed graph with n vertices and at least c(T)n edges and n is sufficiently large, then T is a subgraph of G. In this note we give a proof of this conjecture. In order to prove Theorem 1, we first need to establish the following weaker result.

scholarly journals Vertex-Oriented Hamilton Cycles in Directed Graphs

10.37236/204 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Michael J. Plantholt ◽  
Shailesh K. Tipnis
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Sufficient Conditions ◽  
Hamilton Cycle ◽  
Directed Path ◽  
Hamilton Cycles ◽  
Even Order

Let $D$ be a directed graph of order $n$. An anti-directed Hamilton cycle $H$ in $D$ is a Hamilton cycle in the graph underlying $D$ such that no pair of consecutive arcs in $H$ form a directed path in $D$. We prove that if $D$ is a directed graph with even order $n$ and if the indegree and the outdegree of each vertex of $D$ is at least ${2\over 3}n$ then $D$ contains an anti-directed Hamilton cycle. This improves a bound of Grant. Let $V(D) = P \cup Q$ be a partition of $V(D)$. A $(P,Q)$ vertex-oriented Hamilton cycle in $D$ is a Hamilton cycle $H$ in the graph underlying $D$ such that for each $v \in P$, consecutive arcs of $H$ incident on $v$ do not form a directed path in $D$, and, for each $v \in Q$, consecutive arcs of $H$ incident on $v$ form a directed path in $D$. We give sufficient conditions for the existence of a $(P,Q)$ vertex-oriented Hamilton cycle in $D$ for the cases when $|P| \geq {2\over 3}n$ and when ${1\over 3}n \leq |P| \leq {2\over 3}n$. This sharpens a bound given by Badheka et al.

Antidirected Subtrees of Directed Graphs

1982 ◽  
Vol 25 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Stefan A. Burr
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Personal Communication ◽  
Directed Path ◽  
Directed Tree ◽  
De Bruijn ◽  
Source Sink

The purpose of this paper is to prove the following result:Theorem. Let T be a directed tree with k arcs and with no directed path of length 2. Then if G is any directed graph with n points and at least 4kn arcs, T is a subgraph of G.It would be appropriate to call T an antidirected tree or a source-sink tree, since every point either has all its arcs directed outward or all inward. As N. G. de Bruijn has noted (personal communication), such a linear bound in n cannot hold if T is replaced by any directed graph other than a union of such trees. The above theorem strengthens one of Graham [1], where an implicit bound of c(k)n is obtained, where c(k) is exponentially large. The proof we give here is also shorter. We first give two simple lemmas. Both are essentially due to ErdÄs, but it is not clear where either first appeared. Their proofs are easy, so we give them here for completeness; in neither case do we state quite the best possible result.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

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